Convergence/Divergence of Integrals

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Determine whether the following integral converges or diverges without evaluating it.

[tex]\int_{0}^{1-}\frac{dx}{\sqrt{1-x^4}}[/tex]

Thanks for your help.
 

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  • #2
dextercioby
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That function behaves ~1/x^{2},so it should be integrable in that domain.U have to find a function which is greater than your function,however,it's integral on [0,1] needs to be finite...

[tex] \int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} =\frac{1}{4}\left(\sqrt{\pi}\right)^{3}\frac{\sqrt{2}}{\Gamma^{2}\left(\frac{3}{4}\right)} [/tex]

Daniel.
 
  • #3
Hurkyl
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The problem point is x=1, not x=0. √(1 - x4) behaves like 2√(1-x) there.

P.S. 1/x2 isn't integrable over [0, 1]. :tongue2:
 
  • #4
dextercioby
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I didn't say "integrable".And i didn't say =1/x^{2}...:wink:

Daniel.
 
  • #5
mathwonk
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i believe it certainly converges. now let me think why i say that.

if y^2 = 1-x^4 then 2ydy = -4x^3 dx, so dx/sqrt(1-x^4) = dx/y = -2dy/4x^3. which makes perfectly good sense at x = 1.


huh???

formulas have always seemed like magic to me.
 
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  • #6
dextercioby
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Yeah,but it's still infinite,because your limits of integration will still involve 0...

Daniel.
 
  • #7
saltydog
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dextercioby said:
That function behaves ~1/x^{2},so it should be integrable in that domain.U have to find a function which is greater than your function,however,it's integral on [0,1] needs to be finite...

[tex] \int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} =\frac{1}{4}\left(\sqrt{\pi}\right)^{3}\frac{\sqrt{2}}{\Gamma^{2}\left(\frac{3}{4}\right)} [/tex]

Daniel.
Can you explain how this integral is evaluated in terms of the Gamma function?
 
  • #8
dextercioby
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I dunno.Basically,it returned my a value for the Legendre elliptic integral and then expressed this one in terms of Gamma factor...

Daniel.
 
  • #9
dextercioby
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If u make a substitution,u can transform the integral to

[tex] \int_{1}^{+\infty} \frac{dx}{\sqrt{x^{4}-1}} [/tex]

whose antiderivative,Mathematica finds

Daniel.

P.S.My Maple gave me the results...
 

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